no

It is a good thing to disconnect the negative battery cable when charging, so no damage can occur to the car's computers.

Yes! You can buy trickle chargers that work through the cigarette lighter.

Yes, that is the purpose of a trickle charger. Just make sure it is an automatic charger that comes on when needed and then stops charging when the battery is fully charged. You can leave it on a battery for years if need be.

The trickle, trickle, trickle of the creek water was soothing.